There aren't many unis that offer both, so you're limited in your choices: Swansea Uni, Uni of Wales Trinity St David, Oxford, Liverpool and UCL.
As for Egyptology, a lot of my friends take the course and they enjoy it, mainly because they have a huge interest in the subject to start with (you don't just stumble across such a specialised degree). The one thing I'll note is not many of them seem to have that solid a plan as to what they will do after uni, but maybe that happens in third year. But the uni I'm at, Swansea, has its own Egyptian museum and they do a lot of work in there which they say is great fun and good experience. They also learn Hieroglyphs and ancient Egyptian so that's something to bare in mind incase you don't enjoy languages. Also, as it is such a specialised subject, look through each unis module options. If you are interested in some areas in particular, check if that is an option, wouldn't it be annoying if you never got to touch on your favourite bit?
As for Classics/Classical Civ/Ancient History there are a lot more unis that offer it, so you'll have more choice. They are very versatile degrees, a mix of history, literature, language, religion, art and sometimes archaeology. The way I see them is its the same as if I took English Lit, or History, but a different period of time as they open the same doors and opportunities, plus more.
For Classics, usually you'll be required to study Latin and/or ancient Greek to a high level, although you don't need GCSEs or ALevels in them to apply in most cases, as they will teach you from beginner up. If you hate languages though this is something to remember, and where an Ancient History degree is useful as there are not usually language requirements, but you can still do many of the same modules as Classics. Again, check out the modules because if you are interested in mainly ancient Greek writers, going to a uni that only offers Roman writers isn't ideal.
Also because I've seen this question before, you don't need a Classics qualification, or Latin etc, to get onto a lot of these degree options. This is simply because not many schools offer such options and it isn't fair to students who haven't had the option to ever study it. They usually just want to see you have an interest, can write similar essays and do the work, so History, English and Religious Studies are the sort of ALevels most people on my course have taken, very few have Ancient History, Class Civ or Latin. So don't worry about being behind or not knowing stuff, because for most UK students, the last time they studied the ancients was Roman Britain. And 1st year will usual have an Intro to Ancient Greece module and an Intro to Rome to help you start to piece together the time, useful sources and to figure out what you are really interested in.
If you have any specific questions, please ask, I'm sure we can find an answer.